Irreducibility of automorphic Galois representations of $GL(n)$, $n$ at most $5$

Frank Calegari; Toby Gee

Displaying similar documents to “Irreducibility of automorphic Galois representations of $G L (n)$ , $n$ at most $5$ ”

On the infinite fern of Galois representations of unitary type

Gaëtan Chenevier (2011)

Annales scientifiques de l'École Normale Supérieure

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Let $E$ be a CM number field, $p$ an odd prime totally split in $E$ , and let $X$ be the $p$ -adic analytic space parameterizing the isomorphism classes of $3$ -dimensional semisimple $p$ -adic representations of $Gal (\bar{E} / E)$ satisfying a selfduality condition “of type $U (3)$ ”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in $X$ has dimension at least $3 [E : ℚ]$ . As important steps, and in any rank, we prove that any...

Locally analytic vectors of unitary principal series of ${GL}_{2} (ℚ_{p})$

Ruochuan Liu, Bingyong Xie, Yuancao Zhang (2012)

Annales scientifiques de l'École Normale Supérieure

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The $p$ -adic local Langlands correspondence for ${GL}_{2} (ℚ_{p})$ attaches to any $2$ -dimensional irreducible $p$ -adic representation $V$ of $G_{ℚ_{p}}$ an admissible unitary representation $Π (V)$ of ${GL}_{2} (ℚ_{p})$ . The unitary principal series of ${GL}_{2} (ℚ_{p})$ are those $Π (V)$ corresponding to trianguline representations. In this article, for $p > 2$ , using the machinery of Colmez, we determine the space of locally analytic vectors $Π {(V)}_{an}$ for all non-exceptional unitary principal series $Π (V)$ of ${GL}_{2} (ℚ_{p})$ by proving a conjecture of Emerton.

Dual Blobs and Plancherel Formulas

Ju-Lee Kim (2004)

Bulletin de la Société Mathématique de France

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Let $k$ be a $p$ -adic field. Let $G$ be the group of $k$ -rational points of a connected reductive group $𝖦$ defined over $k$ , and let $𝔤$ be its Lie algebra. Under certain hypotheses on $𝖦$ and $k$ , wethe tempered dual $\hat{G}$ of $G$ via the Plancherel formula on $𝔤$ , using some character expansions. This involves matching spectral decomposition factors of the Plancherel formulas on $𝔤$ and $G$ . As a consequence, we prove that any tempered representation contains a good minimal $𝖪$ -type; we extend this result to irreducible...

When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures

Stéphane R. Louboutin (2020)

Czechoslovak Mathematical Journal

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Let $ε$ be an algebraic unit of the degree $n \geq 3$ . Assume that the extension $ℚ (ε) / ℚ$ is Galois. We would like to determine when the order $ℤ [ε]$ of $ℚ (ε)$ is $Gal (ℚ (ε) / ℚ)$ -invariant, i.e. when the $n$ complex conjugates $ε_{1}, \dots, ε_{n}$ of $ε$ are in $ℤ [ε]$ , which amounts to asking that $ℤ [ε_{1}, \dots, ε_{n}] = ℤ [ε]$ , i.e., that these two orders of $ℚ (ε)$ have the same discriminant. This problem has been solved only for $n = 3$ by using an explicit formula for the discriminant of the order $ℤ [ε_{1}, ε_{2}, ε_{3}]$ . However, there is no known similar formula for $n > 3$ . In the present paper, we put forward and...

On classifying Laguerre polynomials which have Galois group the alternating group

Pradipto Banerjee, Michael Filaseta, Carrie E. Finch, J. Russell Leidy (2013)

Journal de Théorie des Nombres de Bordeaux

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We show that the discriminant of the generalized Laguerre polynomial $L_{n}^{(α)} (x)$ is a non-zero square for some integer pair $(n, α)$ , with $n \geq 1$ , if and only if $(n, α)$ belongs to one of $30$ explicitly given infinite sets of pairs or to an additional finite set of pairs. As a consequence, we obtain new information on when the Galois group of $L_{n}^{(α)} (x)$ over $ℚ$ is the alternating group $A_{n}$ . For example, we establish that for all but finitely many positive integers $n \equiv 2 (mod 4)$ , the only $α$ for which the Galois group of $L_{n}^{(α)} (x)$ over $ℚ$ is $A_{n}$ is...

On critical values of twisted Artin $L$ -functions

Peng-Jie Wong (2017)

Czechoslovak Mathematical Journal

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We give a simple proof that critical values of any Artin $L$ -function attached to a representation $ρ$ with character $χ_{ρ}$ are stable under twisting by a totally even character $χ$ , up to the $dim ρ$ -th power of the Gauss sum related to $χ$ and an element in the field generated by the values of $χ_{ρ}$ and $χ$ over $ℚ$ . This extends a result of Coates and Lichtenbaum as well as the previous work of Ward.

Arithmetic Properties of Generalized Rikuna Polynomials

Z. Chonoles, J. Cullinan, H. Hausman, A.M. Pacelli, S. Pegado, F. Wei (2014)

Publications mathématiques de Besançon

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Fix an integer $ℓ \geq 3$ . Rikuna introduced a polynomial $r (x, t)$ defined over a function field $K (t)$ whose Galois group is cyclic of order $ℓ$ , where $K$ satisfies some mild hypotheses. In this paper we define the family of ${r_{n} (x, t)}_{n \geq 1}$ of degree $ℓ^{n}$ . The $r_{n} (x, t)$ are constructed iteratively from the $r (x, t)$ . We compute the Galois groups of the $r_{n} (x, t)$ for odd $ℓ$ over an arbitrary base field and give applications to arithmetic dynamical systems.

Asymptotic values of modular multiplicities for ${GL}_{2}$

Sandra Rozensztajn (2014)

Journal de Théorie des Nombres de Bordeaux

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We study the irreducible constituents of the reduction modulo $p$ of irreducible algebraic representations $V$ of the group ${Res}_{K / ℚ_{p}} {GL}_{2}$ for $K$ a finite extension of $ℚ_{p}$ . We show that asymptotically, the multiplicity of each constituent depends only on the dimension of $V$ and the central character of its reduction modulo $p$ . As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.

Displaying similar documents to “Irreducibility of automorphic Galois representations of G L ( n ) , n at most 5 ”

Displaying similar documents to “Irreducibility of automorphic Galois representations of $G L (n)$ , $n$ at most $5$ ”